What is rest-pause training?

In one line

Rest-pause is a set method that puts short rests inside a set, so you rest briefly and then keep going with the same weight.

What is rest-pause training?
The method behind 'rest 20 seconds when you can't lift it, then go again'
At the gym you may see someone lift until they cannot do another rep, put the weight down to catch their breath briefly, and then lift the same weight again. This method is called 'rest-pause'. This post explains what rest-pause is and how it differs from similar methods.
In short
A short rest in the middle of a set, then the same weight again
Rest-pause means adding short, preplanned rests inside a set and continuing the reps with the same weight. Normally you rest only between sets, but with rest-pause you also rest within a set.
First: reps, sets, and failure
The moment you cannot lift it again is 'failure'
Lifting a weight once and lowering it is a 'rep', and a group of reps done without rest is a 'set'. The moment during a set when you cannot lift the weight one more time is called 'muscle failure', or just 'failure'.
A common way
Failure, about 20 seconds of rest, failure again
One common way works like this. You lift a weight until failure, put it down, and rest for about 20 seconds. Then you lift the same weight to failure again and rest again. You repeat this until you reach a preplanned number of reps.
Why rest briefly?
To keep lifting the same weight
Right at the moment of failure, you cannot lift the same weight even once more. But after just a few dozen seconds of rest, some strength comes back and you can do a few more reps with the same weight. This rest is usually 10 to 30 seconds, much shorter than the few minutes of rest between sets.
How it differs from a drop set
Lower the weight, or rest briefly?
A drop set continues after failure without rest by lowering the weight. Rest-pause does the opposite: it keeps the weight the same and instead rests briefly before continuing. Both are ways to add more reps after reaching failure.
Why do it?
The same number of reps in less time
Because you rest briefly inside the set instead of resting for minutes between sets, the workout ends sooner. In a study where both groups did the same number of reps as regular sets, a session took about 40% less time. That is why it is described as a method suited to people short on time.
Muscle and strength
No worse than regular sets
In studies where people trained with rest-pause over time, muscle growth and strength gains were no worse than with regular sets. In a study that matched the amount of training, muscle size showed no difference, and squat strength rose slightly more with rest-pause.
Common misconception
Is every rest in the middle of a set rest-pause?
No. Another way to rest within a set is the 'cluster set'. Rest-pause usually rests after getting tired all the way to failure, while a cluster set breaks off and rests before fatigue sets in, to keep each rep fast and strong. The two names are confused often enough that even studies mix them up.
Summary
Short rests inside the set, same weight
Rest-pause means resting briefly in the middle of a set and continuing with the same weight, usually lifting to failure before each rest. It is different from a cluster set, which rests before fatigue sets in. Related terms: drop set, muscle failure, rest time between sets.